Greatest Common Divisor (GCD) of 31 and 140
The greatest common divisor (GCD) of 31 and 140 is 1.
What is the Greatest Common Divisor (GCD)?
The GCD of two integers is the largest positive integer that divides both numbers without leaving a remainder. It is useful in simplifying fractions, finding common factors, and in number theory.
How to Calculate the GCD of 31 and 140?
We use the Euclidean algorithm, which involves the following steps:
- Divide the larger number by the smaller number.
- Replace the larger number with the smaller number and the smaller number with the remainder from the division.
- Repeat this process until the remainder is zero.
- The non-zero remainder just before zero is the GCD.
Step-by-Step Euclidean Algorithm
| Step | Calculation |
|---|---|
| 1 | 31 ÷ 140 = 0 remainder 31 |
| 2 | 140 ÷ 31 = 4 remainder 16 |
| 3 | 31 ÷ 16 = 1 remainder 15 |
| 4 | 16 ÷ 15 = 1 remainder 1 |
| 5 | 15 ÷ 1 = 15 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 108 and 130 | 2 |
| 47 and 169 | 1 |
| 129 and 183 | 3 |
| 200 and 48 | 8 |
| 30 and 55 | 5 |